%Macros
%\newtheorem{lemma}{Lemma}
%\newtheorem{theorem}[lemma]{Theorem}
%\newtheorem{informaltheorem}[lemma]{Informal Theorem}
%\newtheorem{informallemma}[lemma]{Informal Lemma}
%\newtheorem{corollary}[lemma]{Corollary}
%\newtheorem{definition}[lemma]{Definition}
%\newtheorem{proposition}[lemma]{Proposition}
%\newtheorem{question}{Question}
%\newtheorem{problem}{Problem}
%\newtheorem{remark}[lemma]{Remark}
%\newtheorem{claim}{Claim}
%\newtheorem{fact}{Fact}
%\newtheorem{challenge}{Challenge}
%\newtheorem{observation}{Observation}
%\newtheorem{openproblem}{Open Problem}
%\newtheorem{openquestion}{Open question}
\newenvironment{LabeledProof}[1]{\noindent{\bf Proof of #1: }}{\qed}                        

\newenvironment{proofsketch}{\trivlist\item[]\emph{Proof Sketch}:}%
{\unskip\nobreak\hskip 1em plus 1fil\nobreak$\Box$
\parfillskip=0pt%
\endtrivlist}

\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beas}{\begin{eqnarray*}}
\newcommand{\eeas}{\end{eqnarray*}}

\newcommand{\poly}{\mathrm{poly}}
\newcommand{\eps}{\epsilon}
\newcommand{\e}{\epsilon}
\newcommand{\E}{\mbox{E}}
\newcommand{\polylog}{\mathrm{polylog}}
\newcommand{\rob}[1]{\left( #1 \right)} %Round Brackets
\newcommand{\sqb}[1]{\left[ #1 \right]} %square Brackets
\newcommand{\cub}[1]{\left\{ #1 \right\} } %curly brackets
\newcommand{\rb}[1]{\left( #1 \right)} %Round
\newcommand{\abs}[1]{\left| #1 \right|} %| |
\newcommand{\zo}{\{0, 1\}}
\newcommand{\zonzo}{\zo^n \to \zo}
\newcommand{\zokzo}{\zo^k \to \zo}
\newcommand{\zot}{\{0,1,2\}}

\newcommand{\en}[1]{\marginpar{\textbf{#1}}}
\newcommand{\efn}[1]{\footnote{\textbf{#1}}}

\newcommand{\junk}[1]{}
\newcommand{\prob}[1]{\Pr\left[ #1 \right]}
\newcommand{\expt}[1]{\mbox{E}\sqb{#1}}
\newcommand{\expect}[1]{\mbox{E}\left[ #1 \right]}

\newcommand{\md}[1]{\delta_{#1}}                            % min degree
\newcommand{\gf}[1]{G_{#1}}                                     % graph
\newcommand{\nei}[3]{N^{#2}_{#1}\rb{#3}}                % neighbor
\newcommand{\dg}[2]{d_{#1}\rb{#2}}                          % degree
\newcommand{\dgi}[3]{d_{#1}\rb{#2,#3}}                   % induced degree
\newcommand{\prb}[2]{p_{#1,#2}}

%\newcommand{\mindegree}{\delta}

\newcommand{\BfPara}[1]{\noindent {\bf #1}.}

%End macros
